Problem: In triangle $ABC$, $AB = 3$, $BC = 4$, $AC = 5$, and $BD$ is the angle bisector from vertex $B$.  If $BD = k \sqrt{2}$, then find $k$.
Explanation: By Pythagoras, $\angle ABC = 90^\circ$.  Let $P$ and $Q$ be the projections of $D$ onto $BC$ and $AB$, respectively.

[asy]
unitsize(1 cm);

pair A, B, C, D, P, Q;

A = (0,3);
B = (0,0);
C = (4,0);
D = (12/7,12/7);
P = (12/7,0);
Q = (0,12/7);

draw(A--B--C--cycle);
draw(B--D);
draw(P--D--Q);

label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, NE);
label("$P$", P, S);
label("$Q$", Q, W);
label("$x$", (D + P)/2, E);
label("$x$", (D + Q)/2, N);
label("$x$", (B + P)/2, S);
label("$x$", (B + Q)/2, W);
label("$4 - x$", (C + P)/2, S);
label("$3 - x$", (A + Q)/2, W);
[/asy]

We have that $\angle ABC = 90^\circ$ and $\angle PBD = 45^\circ$, so quadrilateral $BPDQ$ is a square.  Let $x$ be the side length of this square.

Then $PC = BC - BP = 4 - x$, and $AQ = AB - QB = 3 - x$.  Triangles $AQD$ and $DPC$ are similar, so \[\frac{AQ}{QD} = \frac{DP}{PC},\]or \[\frac{3 - x}{x} = \frac{x}{4 - x}.\]Solving for $x$, we find $x = 12/7$.  Then $BD = x \sqrt{2} = 12/7 \cdot \sqrt{2}$, so the answer is $\boxed{\frac{12}{7}}$.